Some 42 per cent of respondents surveyed by YouGov do not believe the EU will be prepared to hand back any powers to member states, and another 29 per cent think any concessions won by Britain will only be minor.
Cameron has pledged to renegotiate Britain's terms of membership with the EU before holding a referendum on whether to leave or stay in the block in 2017.
Cameron had waged a very public campaign against Juncker, arguing he was too much of a Brussels insider to push through change, but was outvoted by fellow EU leaders by 26 to 2 at a summit on Friday.
The YouGov poll was conducted before the vote on Thursday but it was clear by then that Cameron had failed to secure enough allies and was heading for defeat.
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Foreign Secretary William Hague said it would be premature to suggest Britain had no hope of securing change.
"It would be far too early to say it doesn't look like we are going to get these reforms," he told BBC television.
"The prime minister acknowledged that the task had got harder, but look at the track record," he said, citing Cameron's success in cutting the EU budget last year.
Health Secretary Jeremy Hunt railed on Saturday against the "cowardice" of other EU leaders who said publicly they would oppose Juncker and then voted for him.
And Hague said: "Everybody can now see in Europe that when the prime minister says he will not back down, he won't back down -- he will stick to his guns."
The YouGov poll found that if there was a referendum tomorrow, 39 per cent would vote to stay in the EU and 37 per cent would vote to leave.